Problem: What is the units digit of the sum of the squares of the first 2007 odd, positive integers?
The units digit of an odd, positive integer can only be 1, 3, 5, 7, or 9. The units digit of the square of an odd, positive integer can only be 1, 9, or 5: $1^2=1$, $3^2=9$, $5^2=25$, $7^2=49$, $9^2=81$. Of every five consecutive odd, positive integers, exactly 2 end in 1 or 9, exactly 2 end in 3 or 7, and exactly 1 ends in 5. Therefore, of the squares of the first $2005=5\cdot401$ odd, positive integers, exactly $\frac{2}{5}\cdot2005=802$ end in 1, exactly $\frac{2}{5}\cdot2005=802$ end in 9, and exactly $\frac{1}{5}\cdot2005=401$ end in 5. The remaining two squares end in 1 ($1^2$) and 9 ($3^2$), respectively. Therefore, the units digit of the sum of the squares of the first 2007 odd, positive integers is the units digit of the sum $802\cdot1+802\cdot9+401\cdot5+1+9$, which is $ \boxed{5}$, the units digit of $2+8+5+0=15$.